Graphs and Combinatorics 6, 1-4 (1990) Combinatorics Summary: Graphs and Combinatorics 6, 1-4 (1990) Graphsand Combinatorics © Springcr-Verlag1990 Transversal Numbers of Uniform Hypergraphs Noga Alon* Department of Mathematics, Sackler Faculty of Exact Sciences,Tel AvivUniversity,Tel Aviv, Israel, and BellCommunications Research, Morristown, NJ 07960,USA Abstract. The transversal number ~(H)of a hypergraph H is the minimum eardinality of a set of verticesthat intersects all edgesofH. For k ~ 1defineck= supz(H)/(ra+ n),where H ranges over all k-uniformhypergraphs with n verticesand medges.Applyingprobabilistic arguments we show that ck= (1 + o(1))~,r. This settles a problem ofTuza. t ~ 1. Introduction The transversal number ~(H) of a hypergraph H is the minimum cardinality of a set of vertices that intersects all edges of H. Let H = (V,E) be a k-uniform hypergraph with n vertices and m edges. Trivially, there exists a positive constant cg (which depends only on k) such that z(H) <_ck(n + m). Tuza [4] proposed the problem of determining or estimating the best possible constants Ckwith the above property. Clearly these constants are given by c~ = sup z(H)/(m + n), where H ranges over all Collections: Mathematics